PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 45, Number 1, July 1974
A SHORT PROOF AND GENERALIZATION OF A
MEASURETHEORETIC DISJOINTIZATIONLEMMA
JOSEPH KUPKA
ABSTRACT.
subfamily
additive
measures
the original
certain
sis
sense,
which
present
measures
present
contains
which
a short
was first
conditions
family
has
The
same
of this
not for a special
as
are,
in a
hypothe-
case
of this
and for which
about
by Rosenthal
of results
a
rethe
proof.
proof of a lemma
obtained
cardinality
continuum
by Rosenthal,
shorter
which
finitely
subfamily
generalized
but
under
of nonnegative,
the
members
obtained
a much
with a number
[2, Lemma
on Banach
of the type of generalization
finitely
spaces.
which
additive
1.1, p. 16] in
Some illustration
may be obtained
is
with the
techniques.
We shall
(see
the
result,
been
yield
general
subfamily
that
for the general
had previously
then provided
this
supported.
paper
connection
such
disjointly
presents
from an infinite
that
and
techniques
This
paper
such
family,
is required
sult
This
may be selected
understand
an ordinal
p. 19]).
As we shall
[l, §4.3,
it will be sufficient
number
rely
for our purposes
i.e.
to define
initial
ordinal
to-one
correspondence
with any of its members.
denote
the
of T,
which
number,
to be a set and not an order
heavily
cardinality
an ordinal
by which
can be put into a one-to-one
the cofinality
such that
note the
of | Y |, by which
I T I contains
power
set
To simplify
a cardinal
which
a cofinal
we have
subset
If T is any set,
with
reworded
let
cardinal
T;
let
cardinal
of cardinality
we mean the set
trivially
to be an
be put into a one-
we mean the unique
correspondence
type
of choice,
number
cannot
we mean the smallest
of V, by which
notation
number
upon the axiom
|Y \
number
cf (V) denote
number
k; and let
of all subsets
Rosenthal's
P(T)
k
de-
of T.
lemma,
which now follows.
1. Lemma. Let T be an infinite
negative
finitely
Received
additive
by the editors
measures
Secondary
Keywords
let \u : x e T] be a family
defined
on all subsets
of T,
of nonand assume
June 25, 1973.
AMS(MOS)subject classifications
04-01;
set,
(1970). Primary 28A10, 28-01, 04A20,
46B99.
and phrases.
Finitely
additive
measures,
cardinality.
Copyright © 1974, American Mathematical Society
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70
A MEASURE THEORETIC DISJOINTIZATION LEMMA
that
sup \u (D:
x e T\
is finite.
X CT such that \X\ = \T\,
Proof.
infinite,
Assume
that,
Then,
for all
e > 0, there
exists
for some
it is a well-known
X
ate pairwise
now select
í > 0, no such
consequence
(For otherwise
p.
disjoint,
an index
set
X exists.
of the axiom
Since
of choice
that
and where
we could
of the Lemma,
If this
procedure
will suffice
: y £ V\ would then satisfy
with
many steps,
cardinal
to assume
number
(i.e.
of this
X
in place
We may
the conclu-
of T, and if it is itera-
then the uniform
sets
boundedness
of the
upon a transfinite
iteration
of the above
only that
that
are
p.
proof.
is hereditary,
pairwise
of p
we
and in fact
it
subset
of
8(x) to be the smallest
disjoint
e1N ) has cardinality
zero,
i.e. that every
of N . Define
every
In place
of p -measure
of
family
of "x-nonnull"
< <5. (Thus,
in Lemma
1,
for all x £ V.)
Assume
let
the generalized
continuum
\N : x e T\ be a family
that there
x e T.
N
S such
which
set,
of sets
a member
Theorem.
and assume
paragraph
is also
we have <5(x)< N
an infinite
lemma is based
N , the collection
a member of N
for all
y eV.
to assumption.)
is repeated
of the first
consider
2.
for all
to be violated.D
the procedure
sets
X = jx
contrary
Our generalization
shall
| = |T|
|T | =
y e V\, where
u x (T ~ X y q ) > e for all x £ Xwy q .
for all y £ Y, an x
£ X
such that
select,
ted for at most finitely
is seen
|X
T is
ynu e T such that
(r ~ X ) < e. The set
sions
a set
and such that px(X ~ ix|) < e for all x e X.
| T x T | [3, (2.2), p. 417], and hence that we have F =\J\Xy:
the
71
exists
Then there
a cardinal
exists
of hereditary
number
a set
hypothesis.
XCT
subsets
k < cf(T)
such
Let
that
Y be
of P(Y),
such
that
8(x) < k
\X\ = | T |, and such
that X ~ ixî e NX ifor all x e X.
Remark.
axioms
In particular,
of set
Proof.
theory
Assume
to
that
well-known
consequence
\G\, where
G denotes
[l, §36.1,
p. 162].
whenever
replace
no such
cf(T)
"< e"
set
by "=0"
X exists.
of the generalized
notation
in Lemma
Since
hypothesis
with domain
we shall
with
treat
the
1.
V is infinite,
continuum
the set of functions
To simplify
> H 2, it is consistent
it is a
that
|T|
k and codomain
the
N
as subsets
=
V
of
P(G).
We now define
hypothesis.
Let
a function
ae
x* e G such
k be fixed,
that
and assume
<5(x*) > k, contrary
by transfinite
induction
to
that
x*(/3) has been defined for all ß < a. Let Ga= ix £ G: x(ß) = x*(ß) fot all
ß < cl\, and observe
that
Ga = \J\Xy:
y £ T\,
where
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Xy = ,x e Ga: x(a)
= y\
72
JOSEPH KUPKA
for all
y e T.
= y. , where
X
The argument
yQ is any member
. Clearly,
3.
then,
Example.
the collection
sors
of x.
easily
Assume
T = c/(D,
subsets
that,
and,
of T which
X C T which
at most a single
In particular,
Ga^
X
e'Nx
for all
x e 1,
are disjoint
8(x) = |x| < | T \ = cf(T)
that any set
2 can contain
that
if T = N.
in general,
and we define
x*(a)
fot all
x £
S(x*) > k, as desired.D
that
Then we have
1 is applicable,
of T such
we have
of those
shown
illustrate
of Lemma
satisfies
let
«x
be
from the predeces-
fot all
x e V, but it is
the conclusions
of Theorem
point.
or N, , a trivial
the
"< e"
elaboration
of Lemma
of this
1 cannot
example
be changed
will
to "= 0"
when cf(F) = N0 or *v
To round
which
out the picture
the hypotheses
4.
under
Proposition.
the formation
exists
a set
a sample
set
2 may be (slightly)
Assume
each
that
of finite
which has cardinality
there
we present
of Theorem
unions,
of the N
and that
of circumstances
of Theorem
it contains
2 is closed
every
subset
< | Y |. Then, provided that 8(x) < cf(T)
XCT
such
that
\X\ = \T\,
under
relaxed.
and such
that
of Y
for all x e T,
X e N
for all
x er.
Proof.
By [3, Theorem
exists a set S C P(D
and suchthat
It is easily
1, p. 45l] (cf. [2, Proposition,
suchthat
\E n F\ < \T\
established
cf(T)
(< |r|)
exist
at least
that,
many sets
|5| > ]T|,
whenever
for all
such that \E\ = |T|
E and
x e T,
E e S such
that
p. 23]), there
F ate distinct
there
must be (strictly)
E éN
one (and in fact > | T | many)
for all E e S,
members
. It follows
X e S such
that
of S.
fewer
that there
X e N
than
must
fot all
x e r. □
5. Example.
the collection
have
S(x) = |T|
Assume that
of those
=cf(D
subsets
|T|
=cf(T),
and, for all x € V, let N
of T which
fot all
have
cardinality
x £ T, whereas
no set
be
< | T |.
Then we
X £ Nx has
|X| = |r|.
REFERENCES
1.
H. Bachmann,
Transfinite
Zahlen,
Ergebnisse
der Math,
und ihrer
Grenzge-
biete, Heft 1, Springer-Verlag, Berlin, 1955. MR 17,1.34.
2.
cations
3.
H. P. Rosenthal,
On relatively
to Banach space
theory, Studia Math. 37 (1970),
W. Sierpinski,
Cardinal
disjoint
and ordinal
families
numbers,
of measures,
13-36.
2nd rev.
ed.,
with
some
Monografie
Mat.,
vol. 34, PWN, Warsaw, 1965. MR 33 #2549.
DEPARTMENT OF MATHEMATICS, MONASHUNIVERSITY, CLAYTON, VICTORIA,
AUSTRALIA
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appli-
MR 42 #5015.